Master Curve Sketching: Enhance Your Calculus I Skills

Question 1

Analyze the concavity of the function f(x) = x^4 - 2x^2.

Accepted Answer

Concave up on (-∞, -1) and (1, ∞)

Answer

Concave up on (-1, 1)

Answer

Concave down on (-∞, -1) and (1, ∞)

Answer

Concave down on (-1, 1)

Question 2

Determine the volume of the solid generated by rotating the region bounded by the function f(x) = x^2 and the x-axis from x = 0 to x = 2 about the x-axis.

Accepted Answer

32π/15

Answer

16π/15

Answer

16π/5

Answer

32π/5

Question 3

Sketch the graph of the function f(x) = (x-2)^2(x+1).

Accepted Answer

Parabola opening upward with vertex (2, -4) and x-intercepts (-1, 0) and (2, 0)

Answer

Parabola opening downward with vertex (2, 4) and x-intercepts (-1, 0) and (2, 0)

Answer

Cubic with a local maximum at x = 2 and a local minimum at x = -1

Answer

Linear function with slope -2 and y-intercept (2, 0)

Question 4

Find the equation of the tangent line to the function f(x) = x^2 - 2x + 1 at the point (1, 0).

Accepted Answer

y = x - 1

Answer

y = x + 1

Answer

y = 2x - 1

Answer

y = -2x + 1

Question 5

Determine whether the graph of the function f(x) = |x-1| has a cusp or a corner point at x = 1.

Accepted Answer

Corner point

Answer

Cusp point

Answer

Neither a cusp nor a corner point

Answer

Both a cusp and a corner point

Question 6

Find the area under the curve of the function f(x) = x^2 + 1 on the interval [0, 2] using integration.

Accepted Answer

7/2

Answer

4/3

Answer

9/2

Answer

5/2

Question 7

Identify the critical points of the function f(x) = x^4 - 4x^2 + 3.

Accepted Answer

x = 0, x = ±2

Answer

x = 0, x = ±1

Answer

x = ±2

Answer

x = ±1

Question 8

Determine the intervals where the function f(x) = (x - 2)^2(x + 3) is increasing.

Accepted Answer

(-∞, -3) U (2, ∞)

Answer

(-3, 2)

Answer

(-∞, 2) U (3, ∞)

Answer

(-3, ∞)

Question 9

Sketch the graph of the function f(x) = |x - 1| + |x + 1|.

Accepted Answer

A V-shaped graph with a vertex at (0, 2)

Answer

An L-shaped graph with a vertex at (0, 2)

Answer

A U-shaped graph with a vertex at (0, 2)

Answer

A straight line with a slope of 1

Question 10

Locate the points of inflection of the function f(x) = x^3 + 3x^2 - 9x + 5.

Accepted Answer

(1, -4)

Answer

(2, 5)

Answer

(-1, 4)

Answer

(-2, -5)

Question 11

Determine whether the function f(x) = x^2e^(-x) is even, odd, or neither.

Accepted Answer

Neither

Answer

Even

Answer

Odd

Question 12

Which of the following steps is NOT essential in curve sketching?

Accepted Answer

Obtaining the second derivative

Answer

Identifying critical points

Answer

Determining intervals of increase and decrease

Answer

Examining concavity

Question 13

Given the derivative f'(x) = (x - 2)(x + 3), what are the critical points of f(x)?

Accepted Answer

x = 2 and x = -3

Answer

x = 0

Answer

x = 2

Answer

x = -3

Question 14

Which of the following is the best description of the graph of the function f(x) = |x - 1| + |x + 1|?

Accepted Answer

A piecewise linear graph with a corner at (0, 2)

Answer

A V-shaped graph with a vertex at (0, 2)

Answer

A U-shaped graph with a vertex at (0, -2)

Answer

A piecewise linear graph with a corner at (0, -2)

Question 15

Determine the derivative of the function f(x) = (x² + 1)⁵.

Accepted Answer

10(x² + 1)⁴ * x

Answer

5(x² + 1)⁴ * x

Answer

(x² + 1)⁵

Answer

5x(x² + 1)⁴

Question 16

Find the inverse function of f(x) = (x - 1)² + 3.

Accepted Answer

f⁻¹(x) = ±√(x - 3) + 1

Answer

f⁻¹(x) = (x - 3)² + 1

Answer

f⁻¹(x) = -√(x - 3) + 1

Answer

The function has no inverse.

Question 17

Determine whether the function f(x) = x⁴ + 2x² - 3 is even, odd, or neither.

Accepted Answer

Even

Answer

Odd

Answer

Neither

Answer

Both even and odd

Question 18

Identify the local maximum of the function h(x) = (x - 1)²(x + 2).

Accepted Answer

(1, 9)

Answer

(2, 0)

Answer

(-2, 9)

Answer

(-1, 0)

Question 19

Find the vertical asymptote(s) of the function g(x) = (x² - 4) / (x - 2).

Accepted Answer

x = 2

Answer

None

Answer

x = -2

Answer

x = -4, 4

Question 20

Identify the equation of the tangent line to the function f(x) = sin(x) at x = π/4.

Accepted Answer

y = √2/2 x + 1/2

Answer

y = √2/2 x - 1/2

Answer

y = -√2/2 x - 1/2

Answer

y = -√2/2 x + 1/2

Question 21

How does an inflection point differ from a critical point?

Accepted Answer

An inflection point changes the concavity of the graph, while a critical point does not.

Answer

A critical point changes the concavity of the graph, while an inflection point does not.

Answer

An inflection point is always a maximum or minimum, while a critical point is not.

Question 22

What is the primary objective of curve sketching in calculus?

Accepted Answer

To visualize and analyze the behavior of functions graphically.

Answer

To solve complex equations numerically.

Answer

To determine the area under curves precisely.

Question 23

Determine whether the graph of the function f(x) = x³ + x² - 2x + 1 has a point of inflection.

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined

Answer

Only if x < 0

Question 24

Find the x-coordinates of the relative extrema of the function f(x) = 2x^3 - 3x^2 - 12x + 5.

Accepted Answer

x = -1 and x = 2

Answer

x = -2 and x = 1

Answer

x = 0 and x = 3

Answer

x = 1 and x = -3

Question 25

Sketch the graph of the function f(x) = (x-2)^2(x+1).

Accepted Answer

A parabola opening upward with a vertex at (-1, 0) and x-intercepts at x = 2 and x = -1

Answer

A line with a slope of -1 and y-intercept of 0

Answer

A parabola opening downward with a vertex at (-1, 0) and x-intercepts at x = 2 and x = -1

Question 26

Identify the concavity of the function f(x) = x^3 + 3x^2 - 9x + 2 on the interval (-∞, -1).

Accepted Answer

Concave down

Answer

Concave up

Answer

Neither concave up nor concave down

Answer

Concave up on the left half and concave down on the right half

Question 27

Find the points of inflection of the function f(x) = x^4 - 4x^3 + 3x^2 + 2.

Accepted Answer

x = 1 and x = 2

Answer

x = -1 and x = 2

Answer

x = 1 and x = -2

Answer

x = 0 and x = 3

Question 28

Determine whether the function f(x) = sin(x) is increasing or decreasing on the interval [0, π/2].

Accepted Answer

Increasing

Answer

Increasing on [0, π/4], decreasing on [π/4, π/2]

Answer

Neither increasing nor decreasing

Answer

Decreasing

Question 29

Sketch the graph of the function f(x) = |x - 1|.

Accepted Answer

A V-shaped graph with a vertex at (1, 0) and two branches: one increasing on (-∞, 1) and one decreasing on (1, ∞)

Answer

A U-shaped graph with a vertex at (1, 0) and two branches: one increasing on (-∞, 1) and one increasing on (1, ∞)

Question 30

Find the equation of the tangent line to the curve y = x^3 at the point (1, 1).

Accepted Answer

y = 3x - 2

Answer

y = x + 2

Answer

y = x^3 - 2

Question 31

Determine the critical points of the function f(x) = x^3 - 3x^2.

Accepted Answer

x = 0 and x = 2

Answer

x = 0 only

Answer

x = 1 and x = 3

Answer

x = -1 and x = 1

Question 32

Analyze the behavior of the function f(x) = x^4 - 4x^3 at x = 0. What graphical feature is present at this point?

Accepted Answer

The function has a horizontal tangent line at x = 0.

Answer

The function has a cusp at x = 0.

Answer

The function has a vertical tangent line at x = 0.

Answer

The function has a local minimum at x = 0.

Question 33

Identify the intervals where the function f(x) = x^3 - 6x^2 + 9x is strictly increasing.

Accepted Answer

x < 1 and x > 3

Answer

x < 1

Answer

x < 0 and x > 2

Answer

1 < x < 3

Question 34

At what x-value does the function f(x) = x^3 exhibit a point of inflection?

Accepted Answer

x = 0

Answer

x = 1

Answer

x = -1

Answer

The function has no point of inflection.

Question 35

Which of the following descriptions accurately represents a function with a vertical asymptote at x = 2 and a horizontal asymptote at y = 1?

Accepted Answer

The function approaches infinity as x approaches 2, and approaches 1 as x approaches positive or negative infinity.

Answer

The function has a hole at x=2 and approaches 1 as x approaches infinity.

Answer

The function approaches infinity as x approaches both 2 and negative infinity.

Answer

The function approaches 1 as x approaches 2, and approaches infinity as x approaches positive or negative infinity.

Question 36

Given f'(x) = 2x - 3, determine the equation of the tangent line to the graph of f(x) at the point (2, 1).

Accepted Answer

y = x - 1

Answer

y = 2x - 3

Answer

y = -x + 3

Answer

y = 3x - 5

Question 37

Which of the following functions exhibits concave down behavior on the interval (0, 1)?

Accepted Answer

f(x) = -x^2 + 2x

Answer

f(x) = x^3 - 3x

Answer

f(x) = x^4 - 4x^3

Answer

f(x) = ln(x)

Question 38

Find the absolute maximum value of the function f(x) = x^3 - 3x^2 + 2 on the closed interval [0, 2].

Accepted Answer

2

Answer

-1

Answer

0

Answer

1

Question 39

Which function possesses a vertical asymptote at x = 0?

Accepted Answer

f(x) = 1/x

Answer

f(x) = e^x

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Question 40

The graph of f(x) displays a horizontal tangent line at x = 3. What can be concluded about the value of f'(3)?

Accepted Answer

f'(3) = 0

Answer

f'(3) = 1

Answer

You cannot conclude anything about f'(3).

Answer

f'(3) is undefined

Question 41

Find the critical points of the function  *f(x) = x³ - 3x² + 2x*.

Accepted Answer

x = 1 and x = 2/3

Answer

x = -1 and x = 2

Answer

x = 0 and x = 1

Answer

x = 1 and x = -2/3

Question 42

At *x = 0*, the function *f(x) = x⁴ - 2x² + 1* has a:

Accepted Answer

Local minimum

Answer

Saddle point

Answer

Local maximum

Answer

Inflection point

Question 43

The function *f(x) = x³ - 3x* has an inflection point at:

Accepted Answer

x = 0

Answer

x = 3

Answer

x = 1

Answer

x = -1

Question 44

What is the horizontal asymptote of the function *f(x) = (x² + 1) / (x² - 1)*?

Accepted Answer

y = 1

Answer

x = 1

Answer

x = -1

Answer

y = 0

Question 45

The function *f(x) = e^x* has:

Accepted Answer

No horizontal asymptote

Answer

A horizontal asymptote at y = 0

Answer

A horizontal asymptote at y = 1

Answer

A horizontal asymptote at y = -1

Question 46

Which of the following functions has a vertical asymptote at *x = 2*?

Accepted Answer

f(x) = 1 / (x - 2)

Answer

f(x) = (x - 2) / (x² + 1)

Answer

f(x) = 1 / (x² - 4)

Answer

f(x) = x² - 4

Question 47

The function *f(x) = sin(x)* is:

Accepted Answer

Periodic

Answer

Linear

Answer

Odd

Answer

Even

Question 48

The graph of *f(x) = x²* is symmetric about:

Accepted Answer

The y-axis

Answer

The line y = x

Answer

The x-axis

Answer

The origin

Question 49

Which of the following is NOT a step in curve sketching?

Accepted Answer

Finding the domain of the function

Answer

Determining the concavity of the function

Answer

Analyzing the behavior of the function near its asymptotes

Answer

Finding the critical points

Question 50

The graph of a function with a positive first derivative and a negative second derivative is:

Accepted Answer

Increasing and concave down

Answer

Increasing and concave up

Answer

Decreasing and concave up

Answer

Decreasing and concave down